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<b>let </b><font color="Maroon" title="c1">seq'</font>, <font color="Maroon" title="c2">seq</font> be   <a href="seq_1.html#NM1" title="SEQ_1:NM.1">Real_Sequence</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c1">seq'</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp; <font color="Maroon" title="c2">seq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a> &amp;  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  &amp; <font color="Maroon" title="c2">seq</font> <a href="seq_1.html#NR1" title="SEQ_1:NR.1">is_not_0</a>  implies  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq'</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c1">seq'</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span> )</span><br/>



<b>assume </b><b>that </b><br/><a NAME="E1:30"/><i><font color="Green" title="E25">A1</font></i>: 
<font color="Maroon" title="c1">seq'</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>and </b><br/><a NAME="E2:30"/><i><font color="Green" title="E26">A2</font></i>: 
<font color="Maroon" title="c2">seq</font> is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 <b>and </b><br/><a NAME="E3:30"/><i><font color="Green" title="E27">A3</font></i>: 
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 <b>and </b><br/><a NAME="E4:30"/><i><font color="Green" title="E28">A4</font></i>: 
<font color="Maroon" title="c2">seq</font> <a href="seq_1.html#NR1" title="SEQ_1:NR.1">is_not_0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq'</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c1">seq'</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span></span><br/>

<a NAME="E5:30"/>
<font color="Maroon" title="c2">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a>  is <a href="seq_2.html#V4" title="SEQ_2:attr.4">convergent</a>
 
<b>by </b><i><a class="txt" href="seq_2.html#E2:30"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="seq_2.html#E3:30"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="seq_2.html#E4:30"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="seq_2.html#T35" target="_self" title="SEQ_2:th.35">Th35</a></i>;<br/>
<b>then </b> <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq'</font> <a href="valued_1.html#K20" title="VALUED_1:func.20">(#)</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c1">seq'</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p2">(<span class="default"><font color="Maroon" title="c2">seq</font> <a href="valued_1.html#K37" title="VALUED_1:func.37">"</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="seq_2.html#E1:30"><i><font color="Green" title="E25">A1</font></i></a>, <a class="ref" href="seq_2.html#T29" target="_self" title="SEQ_2:th.29">Th29</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c1">seq'</font></span>)</span> <a href="real_1.html#K8" title="REAL_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="real_1.html#K2" title="REAL_1:func.2">"</a> </span>)</span>
<b>by </b><i><a class="txt" href="seq_2.html#E2:30"><i><font color="Green" title="E26">A2</font></i></a>, <a class="txt" href="seq_2.html#E3:30"><i><font color="Green" title="E27">A3</font></i></a>, <a class="txt" href="seq_2.html#E4:30"><i><font color="Green" title="E28">A4</font></i></a>, <a class="ref" href="seq_2.html#T36" target="_self" title="SEQ_2:th.36">Th36</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c1">seq'</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span>
<b>by </b><i><a class="ref" href="xcmplx_0.html#D9" title="XCMPLX_0:def.9">XCMPLX_0:def 9</a></i>
;<br/>
<b>hence </b><a NAME="E7:30"/>
 <a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <span class="p1">(<span class="default"><font color="Maroon" title="c1">seq'</font> <a href="valued_1.html#K52" title="VALUED_1:func.52">/"</a> <font color="Maroon" title="c2">seq</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c1">seq'</font></span>)</span> <a href="real_1.html#K10" title="REAL_1:func.10">/</a> <span class="p1">(<span class="default"><a href="seq_2.html#K2" title="SEQ_2:func.2">lim</a> <font color="Maroon" title="c2">seq</font></span>)</span>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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